Research Summary
Primes in Geometric-Arithmetic Progression

A geometric-arithmetic progression of primes is a set of k primes
(denoted by GAP-k) of the form p1*r j + j*d
for fixed p1, r and d and consecutive j,
from j = 0 to k - 1.
i.e, {p1, p1*r + d, p1*r 2 + 2 d,
p1* r 3 + 3 d, ...}.
For example 3, 17, 79 is a 3-term geometric-arithmetic progression
(i.e, a GAP-3) with a = p1 = 3, r = 5 and d = 2.
A GAP-k is said to be minimal if the minimal start p1 and
the minimal ratio r are equal, i.e, p1 = r = p, where p
is the smallest prime ≥ k.
Such GAPs have the form p*p j + j*d.
Minimal GAPs with different differences, d do exist. For example, the minimal GAP-5
(p1 = r = 5) has the
possible differences, 84, 114, 138, 168, ... (see the Sequence A209204)
and the minimal
GAP-6 (p1 = r = 7) has the possible differences,
144, 1494, 1740, 2040, .... (see the Sequence A209205).
A minimal GAP-k is further said to be absolutely minimal if the difference d
is minimum.
All the GAPs up to k = 12 in Table-1 are absolutely minimal.
Table-2 has the integer sequences for the differences
corresponding to the minimal GAPs up to k = 12.
Table-3 has the miscellaneous examples for the non-minimal GAPs.
The following article gives the conditions under which, a GAP-k is a
set of k primes in geometric-arithmetic progression.
Computational data was obtained using initially the
Microsoft EXCEL (up to GAP-6 in Table-1)
and then the versatile MATHEMATICA.

Quadricmeter is the instrument devised to identify (distinguish) and measure the various
parameters (axis, foci, latera recta, directrix, etc.,) completely characterizing the important
class of surfaces known as the quadratic surfaces. Quadratic surfaces (also known as quadrics)
include a wide range of commonly encountered surfaces including, cone, cylinder, ellipsoid,
elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder,
hyperbolic paraboloid, paraboloid, sphere, and spheroid. Quadricmeter is a generalized form of
the conventional spherometer and the lesser known cylindrometer (also known as the "Cylindro-Spherometer"
and "Sphero-Cylindrometer").
With a conventional spherometer it was possible only to measure the radii of spherical surfaces.
Cylindrometer can measure the radii of curvature of a cylindrical surface in addition to the spherical
surface. In both the spherometer and the cylindrometer one assumes the surface to be either spherical
or cylindrical respectively. In the case of the quadricmeter, there are no such assumptions.